1. Field of the Invention
The invention relates generally to complex analog-to-digital converters, and more specifically to methods and apparatus for complex sigma-delta modulation.
2. Discussion of the Related Art
Analog-to-digital converters (ADCs) are used to convert analog information to digital information so that signal processing may be accomplished in the digital domain. In particular, sigma-delta ADCs are useful in such applications. Sigma-delta ADCs convert incoming analog signals in a particular frequency span of interest into a high-rate (oversampled), low resolution (one-bit) digital output data stream. The sigma-delta approach to analog-to-digital conversion is well-known for its superior linearity and anti-aliasing performance compared to traditional ADC conversion approaches with lower sampling rates.
In order to maintain the full performance of sigma-delta conversion, it is desirable to implement a “complex” converter, which may be thought of as converting a pair of input signals into streams of digital output values, one such stream representing the “real” or “in-phase” (I) component of the signal, and the other such stream representing the “imaginary” or “quadrature” (Q) component of the signal. It is convenient and common to represent the two output data streams I and Q as a single complex data stream I+jQ, where j is a symbol representing the square root of −1.
The advantages of sigma-delta modulators come at some expense. For example, the quantization of the signal produces noise in the output data stream, known as quantization noise. An important job of a sigma-delta converter is to “shape” this quantization noise out of the frequency range which contains the desired signal, so that subsequent digital filtering operations may recover the desired signal without corruption. In a subsequent stage, this out-of-band quantization noise may be eliminated by means of a filter. In the case of a low-pass sigma-delta modulator, the band of interest spans a frequency range centered around DC, as shown in FIG. 1A, whereas in a bandpass sigma-delta modulator, the center frequency is shifted to a higher frequency, as shown in FIG. 1B.
Two basic possibilities to improve the performance of sigma-delta modulators are the use of a higher-order modulator, or the use of a multi-bit quantizer. These approaches are not necessarily the most effective solutions. The former leads to system instability and latter may cause non-linearity. Cascading of low-order single-bit modulators has been proven to be an efficient way to achieve a higher performance without facing the above-mentioned problems. Cascaded modulators require a digital noise cancellation circuit to remove the quantization noise introduced by the first stages. Consequently, the output quantization noise will be ideally due to the very last stage of the modulator.
As may be seen in references FIGS. 1A and 1B, quantization noise 101, 102+ and 102− and the desired signal 105, 106+ and 106− all remain symmetric with respect to the vertical axes 103, 104 for both the low-pass and bandpass modulators. This way of shaping the quantization noise is wasteful when one side of the spectrum, for example, positive frequencies, provides all the information carried by the signal. For instance, the quanrature bandpass signal shown in FIG. 2 exhibits such a property. This property has been the main motivation for using complex sigma-delta modulators for quanrature bandpass signals. A complex sigma-delta modulator may be implemented using two real modulators with the interconnections between them in such a way that the output complex signal, yr+jyi, exhibits an asymmetric spectrum for quantization noise.
However, it is not possible to use the same principle for baseband signals because the complex signal I+jQ has spectral content at both positive and negative frequencies. For this reason, only real sigma-delta modulators with a symmetric noise shaping characteristic have been used for direct conversion systems, and two real sigma-delta modulators have been required to process the in-phase and quadrature components.
A single complex modulator is far more efficient in terms of noise shaping than two real modulators operating separately with xr=I and xI=Q. In other words, for a given number of integrators, a complex sigma-delta modulator provides a better signal-to-noise (SNR) ratio. Alternatively, for a given SNR, a complex modulator requires a smaller number of integrators. This, in turn, translates into a smaller chip area and lower power consumption. The main issue with both cascade and complex modulators is their sensitivity to variation of coefficients. Inaccuracy of the coefficients in a complex modulator degrades the quality of noise shaping and causes image leakage. In a cascade structure, mismatch between the coefficients of the modulator and the coefficients of the digital noise cancellation circuit limits the achievable SNR.